Find the distance between the parallel lines
12x + 5y = 7 and 12x + 5y + 7 = 0.

Find the acute angle between the pair of lines given by 2x²- 5xy - 7y² = 0.

Find the path traced out by the point \((ct,\frac{c}{t})\) , here t ≠ 0 is the parameter and c is a constant.

Find the equation of the line through the point of intersection of the line 5x - 6y = 1 and 3x + 2y + 5 = 0 and cutting off equal intercepts on the coordinate axis.

Find the equation of the line through (1, 2) and which is perpendicular to the line joining (2, -3) (-1, 5)..

Find the equation of the straight line through the intersection of 5x - 6y = 1 and 3x + 2y + 5 = 0 and perpendicular to the straight line 3x - 5y + 11 =0.

Show that x² - y² + x - 3y - 2 = 0 represents a pair of straight lines. Find also angle between the lines.

If the equation 12x² - 10xy + 2y² + 14x - 5y + k = 0 represents a pair of straight lines, find k, find separate equation and also angle between them.

Find the angle between the lines 3x² + 10xy + 8y² + 14x + 22y + 15 = 0.

Evaluate: \(\underset { n\rightarrow \infty }{ lim } \cfrac { 1+2+3+...+n }{ { n }^{ 2 } } \)